$2.$ Consider the mechanism design example discussed in class in which there are two risk$-$neutral bidders with independent private values uniformly distributed on $[1,2]$ and $[0,3],$ respectively. The risk$-$neutral seller$$s value from retaining the object is zero. $($a$)$ Derive the expected payment functions ei$($ti$)$ for i $=1,2.$ Explain. $($b$)$ Can you think of an auction whose BNE implements $($i$.$e$.$ finds the same e and p functions as$)$ the optimal direct incentive compatible mechanism? Explain.